We introduce the notion of an L-massive subset in a noncompact Riemannian manifold and study properties of such subsets. It is proved that a semilinear elliptic equation on a noncompact Riemannian manifold has a nontrivial bounded solution if and only if there exists an L-massive subset in this manifold. A similar assertion is also proved for solutions of semilinear equations with finite energy integral.

中文翻译：

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关于非紧致黎曼流形的容量特性

我们介绍了非紧黎曼流形中的L-大规模子集的概念，并研究了此类子集的性质。证明了，当且仅当在该流形上存在一个L-大规模子集时，一个非紧黎曼流形上的一个半线性椭圆方程具有一个非平凡的有界解 。对于具有有限能量积分的半线性方程组的解，也证明了类似的断言。